Taylor’s formula with n = 1 and a = 0 gives the linearization of a function at x = 0. With n = 2 and n = 3 we obtain the standard

quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions:

a. For what values of x can the function be replaced by each approximation with an error less than

b. What is the maximum error we could expect if we replace the function by each approximation over the specified interval?

Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals in Exercises

Step 1: Plot the function over the specified interval.

Step 2: Find the Taylor polynomials P1(x), P2(x), and P3(x) at x = 0.

Step 3: Calculate (n + 1)st the derivative ƒ(n+1)(c), associated with the remainder term for each Taylor polynomial. Plot the derivative as a function of c over the specified interval and estimate its maximum absolute value, M.

Step 4: Calculate the remainder for each polynomial. Using the estimate M from Step 3 in place of ƒ(n+1)(c), plot Rn(x) over the specified interval. Then estimate the values of x that answer question (a).

Step 5: Compare your estimated error with the actual error by plotting En(x) over the specified interval. This will help answer question (b).

Step 6: Graph the function and its three Taylor approximations together. Discuss the graphs in relation to the information discovered in Steps 4 and 5.